added lec 19

content/Convolution (lec 19).md

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+# Convolution
+A convolution is an operation of function, we take two functions, convolute them and get a new function.
+Definition of convolution between f and g:
+$$(f*g)(t):=\int _{0} ^t f(t-v)g(v)\, dv$$
+property 1) $f*g=g*f$
+proof:
+$f*g=\int _{0} ^t f(t-v)g(v)\, \underset{ t-v=u }{ dv }=-\int _{t} ^0 f(u)g(t-u) \, du$
+$=\int _{0} ^t g(t-u)f(u)\, du=g*f \quad \Box$
+
+property 2) $(f+g)*h=f*h+g*h$
+property 3) $(f*g)*h=f*(g*h)$
+property 4) $f*0=0$
+property 5) $\mathcal{L}\{f*g\}=F(s)G(s)$
+he will see us tomorrow at 10oclock. ;)
+#end of lec 19

content/Periodic functions (lec 19).md

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+#start of lec 19
+This lecture we will learn about periodic functions, specifically, non-sinusoidal periodic functions.
+# Periodic function
+Definition:
+$f$ is periodic with period $T \quad (T>0)$ if: 
+$$f(t)=f(t+T), \quad \forall\ t\in \mathbb{R}$$
+![[Drawing 2023-10-20 13.06.35.excalidraw]]
+We will now compute laplace transforms of these periodic functions. Computing DE's containing these periodic functions using something like #voparam would not be easy.
+If we take the windowed version of the function (one period, where everywhere else is 0, ie:)
+
+$f_{T}(t)=\begin{cases}f(t)\ ,\  & 0\leq t\leq T \\0\ ,\ & \text{otherwise}\end{cases}$
+
+we can "glue together" many of these windows together to rebuild our $f(t)$, like this:
+$f(t)=f_{T}(t)+f_{T}(t-T)u(t-T)+f_{T}(t-2T)u(t-2T)+\dots$
+$\mathcal{L}\{f\}=\mathcal{L}\{f_{T}\}+\mathcal{L}\{f(t-T)u(t-T)\}+\dots$
+recall the formula from last lec: $\mathcal{L}\{u(t-a)f(t-a)\}=e^{-as}F(s)$
+then:
+$\mathcal{L}\{f\}=\mathcal{L}\{f_{T}\}(1+e^{-TS}+e^{-2TS}+e^{-3TS}+\dots)$
+$\mathcal{L}\{f\}=\mathcal{L}\{f_{T}\}(1+e^{-TS}+(e^{-TS})^{2}+(e^{-TS})^{3}+\dots)$
+This is a geometric series! $1+r+r^2+\dots$ 
+Geometric series are convergent when $|r|<1$
+and equal to: $\frac{1}{1-r}$
+in this case, $r=e^{-Ts}$
+so: 
+$$\mathcal{L}\{f\}=\mathcal{L}\{f_{T}\} \frac{1}{1-e^{-Ts}}$$
+handy formula! ^ will be used again.
+#ex
+imagine another function: (image is of a square wave with a period of 2a, oscillates between 1 and 0, starts at 1 when t=0.)
+![[Drawing 2023-10-20 13.27.58.excalidraw]]
+$\mathcal{L}\{f\}=\mathcal{L}\{f_{2a}\} \frac{1}{1-e^{-2as}}$
+$f_{2a}=u(t)-u(t-a)$ (this is the first period piece)
+$\implies \mathcal{L}\{f_{2a}\}=\mathcal{L}\{u(t)\}-\mathcal{L}\{u(t-a)\}=\frac{1}{s}- \frac{e^{-as}}{s}$
+plug back in:
+$\mathcal{L}\{f\}=\mathcal{L}\{f_{2a}\} \frac{1}{1-e^{-2as}}=\frac{1}{s}\cancel{ (1-e^{-as}) } \frac{1}{\cancel{ (1-e^{-as}) }(1+e^{-as})}$
+$$\mathcal{L}\{f\}=\frac{1}{s(1+e^{-as})}$$
+
+#ex 
+$y''+3y'+2y=f(t)$ where $f(t)$ is from the previous example
+$y(0)=y'(0)=0,\ a=1$ (a is width of 1/2 period in the function f(t))
+$s^2Y+3sY+2Y=\mathcal{L}\{f\}= \frac{1}{s(1+e^{-s})}$
+$Y(s)=\frac{1}{s(s+1)(s+2)} \frac{1}{1+e^{-s}}$
+$=F(s) \frac{1}{1+e^{-s}}$
+
+$\mathcal{L}^{-1}\{F\}=\mathcal{L}^{-1}\{\frac{1}{2} \frac{1}{s}+\frac{1}{2} \frac{1}{s+2}-\frac{1}{s+1}\}$
+$=\frac{1}{2}+\frac{1}{2}e^{-2t}-e^{-t}$
+$y(t)= \dots$
+we want to use formula from earlier
+so we need to change  $F(s) \frac{1}{1+e^{-s}}$
+to: $F(s) \frac{1e^{-s}}{1-e^{-2s}}$
+$y(t)=p(t)$, periodic with period of 2 ($T=2$)
+
+$$f_{2a}(t)=\mathcal{L}^{-1}\{F(s)-F(s)e^{-s}\}=\frac{1}{2}+\frac{1}{2}e^{-2t}-e^{-t}-\left( \frac{1}{2}+\frac{1}{2} e^{-2(t-1)}-e^{-(t-1)})u(t-1 \right)$$
+
+

content/_index.md

@@ -19,8 +19,8 @@ Good luck on midterms! <3 -Oct 18 2023
 [Laplace transform (lec 14-16)](laplace-transform-lec-14-16.html) (raw notes, not reviewed or revised yet.)
 [Solving IVP's using Laplace transform (lec 17-18)](solving-ivps-using-laplace-transform-lec-17-18.html) (raw notes, not reviewed or revised yet.)
 [(Heaviside) Unit step function (lec 18)](heaviside-unit-step-function-lec-18.html) (raw notes, not reviewed or revised yet.)
-
+[Periodic functions (lec 19)](periodic-functions-lec-19.html) (raw notes, not reviewed or revised yet.)
-
+[Convolution (lec 19)](convolution-lec-19.html)
 </br>
-[How to solve any DE, a flow chart](Solve-any-DE.png)
+[How to solve any DE, a flow chart](Solve-any-DE.png) (Last updated Oct 1st, needs revision. But it gives a nice overview.)
 </br>